^{1}, Gregory Toepperwein

^{2}, George J. Papakonstantopoulos

^{3}, Jean-Louis Barrat

^{4}and Juan J. de Pablo

^{2,a)}

### Abstract

Polymernanocomposites have been widely studied in efforts to engineer materials with mechanical properties superior to those of the pure polymer, but the molecular origins of the sought-after improved properties have remained elusive. An ideal polymernanocomposite model has been conceived in which the nanoparticles are dispersed throughout the polymeric matrix. A detailed examination of topological constraints (or entanglements) in a nanocomposite glass provides new insights into the molecular origin of the improved properties in polymernanocomposites by revealing that the nanoparticles impart significant enhancements to the entanglement network. Nanoparticles are found to serve as entanglement attractors, particularly at large deformations, altering the topological constraint network that arises in the composite material.

We acknowledge the National Science Foundation for funding through the MRSEC on nanostructured interfaces at the University of Wisconsin and the NIRT Grant No. CTS-0506840.

I. INTRODUCTION

II. METHODS

A. Molecular model

B. Primitive path analysis

III. RESULTS

IV. SUMMARY

### Key Topics

- Polymers
- 81.0
- Nanoparticles
- 37.0
- Nanocomposites
- 27.0
- Elastic moduli
- 9.0
- Reinforced polymers
- 7.0

## Figures

Strain response of the pure polymer (○) and the nanocomposite (◻) to both tensile and compressive stresses of . The positive strain responses correspond to tensile deformations and the negative strain responses to compressive deformations. We find that in all cases the presence of the nanoparticles suppresses the creep response of the glass, and the differences become more pronounced at longer times.

Strain response of the pure polymer (○) and the nanocomposite (◻) to both tensile and compressive stresses of . The positive strain responses correspond to tensile deformations and the negative strain responses to compressive deformations. We find that in all cases the presence of the nanoparticles suppresses the creep response of the glass, and the differences become more pronounced at longer times.

(a) Average nonaffine displacement per particle plotted against the strain for the pure polymer (○), the nanocomposite (◻), and the center of mass of the nanoparticles in the nanocomposite system (◇). Error bars represent the standard relative error around the average of the three configurations for each system. The dashed, dotted, and dotted-dashed lines with , (△), and (▽) symbols each represents the nonaffine displacements for individual particles. In all cases, the uncertainties are approximately the size of the symbols. (b) and (c) show the distribution of nanoparticles in the nanocomposite system immediately after application of stress (b) and when the stress was removed (c). Also included in (b) and (c) are the particles with a nonaffine displacement larger than 1.0, which tend to concentrate away from the nanoparticle surfaces.

(a) Average nonaffine displacement per particle plotted against the strain for the pure polymer (○), the nanocomposite (◻), and the center of mass of the nanoparticles in the nanocomposite system (◇). Error bars represent the standard relative error around the average of the three configurations for each system. The dashed, dotted, and dotted-dashed lines with , (△), and (▽) symbols each represents the nonaffine displacements for individual particles. In all cases, the uncertainties are approximately the size of the symbols. (b) and (c) show the distribution of nanoparticles in the nanocomposite system immediately after application of stress (b) and when the stress was removed (c). Also included in (b) and (c) are the particles with a nonaffine displacement larger than 1.0, which tend to concentrate away from the nanoparticle surfaces.

Cumulative probability that all of the entanglement junctions are farther than a given distance from a particle (○). For comparison, we also show the cumulative probability that all of the particles are farther than a given distance from a central particle (◻), as well as the cumulative probability that all of the entanglements are farther than a given distances from a central entanglement (△). This plot demonstrates that the entanglements due to polymer chain crossings are typically closer than the next-nearest nanoparticle. The dashed lines indicate the same probability distributions after deformation, and we find minimal changes in the distribution of the chain entanglements.

Cumulative probability that all of the entanglement junctions are farther than a given distance from a particle (○). For comparison, we also show the cumulative probability that all of the particles are farther than a given distance from a central particle (◻), as well as the cumulative probability that all of the entanglements are farther than a given distances from a central entanglement (△). This plot demonstrates that the entanglements due to polymer chain crossings are typically closer than the next-nearest nanoparticle. The dashed lines indicate the same probability distributions after deformation, and we find minimal changes in the distribution of the chain entanglements.

Images highlighting the same nanoparticle and all of the primitive paths contacting this particle before the deformation (top) and after experiencing tensile stress for time steps and reaching a strain of (bottom). The thin lines represent the entanglement network for the remaining polymer (only half of the primitive paths are shown for clarity).

Images highlighting the same nanoparticle and all of the primitive paths contacting this particle before the deformation (top) and after experiencing tensile stress for time steps and reaching a strain of (bottom). The thin lines represent the entanglement network for the remaining polymer (only half of the primitive paths are shown for clarity).

(a) Probability that a particle has a given number of contacts in the initial state (○), after compressive deformation (◇), and after tensile deformation (◻). The uncertainties are approximately the size of the symbols. (b) The total number of primitive path contacts per nanoparticle as the system deforms in tension (solid line) or compression (dashed line). (c) Number of primitive path contacts plotted against the instantaneous strain for three chosen particles as the nanocomposite system deforms. The nonaffine displacements for these three particles are shown in Fig. 2(a), and the particles with fewer primitive path contacts tend to have larger nonaffine displacements, while the nanoparticles that are trapped by entanglements tend to move in an affine manner.

(a) Probability that a particle has a given number of contacts in the initial state (○), after compressive deformation (◇), and after tensile deformation (◻). The uncertainties are approximately the size of the symbols. (b) The total number of primitive path contacts per nanoparticle as the system deforms in tension (solid line) or compression (dashed line). (c) Number of primitive path contacts plotted against the instantaneous strain for three chosen particles as the nanocomposite system deforms. The nonaffine displacements for these three particles are shown in Fig. 2(a), and the particles with fewer primitive path contacts tend to have larger nonaffine displacements, while the nanoparticles that are trapped by entanglements tend to move in an affine manner.

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